For a mass-spring system with angular frequency ω, what is the period T of the motion?

For a mass-spring system with angular frequency ω, what is the period T of the motion?

• A. T = 2π/ω
• B. T = ω
• C. T = π/ω
• D. T = 1/ω

Answer: (A)

In simple harmonic motion, the displacement follows x(t) = A cos(ωt + φ), so the phase advances at a rate of ω radians per second. A full cycle corresponds to a phase change of 2π radians, which means the time for one complete oscillation is T = 2π/ω. This also matches the relation between angular frequency and ordinary frequency, f = ω/(2π), since T = 1/f. The other options don’t fit: replacing ω with ω in the denominator would give units of 1/s, not seconds, so that isn’t a period. Using π/ω would represent a half-cycle, not a full cycle. Using 1/ω would yield seconds per radian, not seconds per cycle.

In simple harmonic motion, the displacement follows x(t) = A cos(ωt + φ), so the phase advances at a rate of ω radians per second. A full cycle corresponds to a phase change of 2π radians, which means the time for one complete oscillation is T = 2π/ω. This also matches the relation between angular frequency and ordinary frequency, f = ω/(2π), since T = 1/f.

The other options don’t fit: replacing ω with ω in the denominator would give units of 1/s, not seconds, so that isn’t a period. Using π/ω would represent a half-cycle, not a full cycle. Using 1/ω would yield seconds per radian, not seconds per cycle.